Some Applications of Wagner's Weighted Subgraph Counting Polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

The m-Path Cover Polynomial of a Graph and a Model for General Coefficient Linear Recurrences

An m-path cover Γ={Pℓ1,Pℓ2,…,Pℓr} of a simple graph G is a set of vertex disjoint paths of G, each with ℓk≤m vertices, that span G. With every Pℓ we associate a weight, ω(Pℓ), and define the weight of Γ to be ω(Γ)=∏k=1r‍ω(Pℓk). The m-path cover polynomial of G is then defined as ℙm(G)=∑Γ‍ω(Γ), where the sum is taken over all m-path covers Γ of G. This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the m-path cover polynomial of a weighted path P(m-1,n) and find the (m+1)-term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the m-path cover polynomial of a suitably weighted cycle C(n). A directed graph, T(m), the edge-weighted m-trellis, is introduced and so a third way to generate the solutions to the above (m+1)-term recurrence is presented. We also give a model for general-term linear recurrences and time-dependent Markov chains.

Graphs Whose Certain Polynomials Have Few Distinct Roots

Let G=(V,E) be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of G. Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded.

Zhang – Zhang Polynomial of Multiple Linear Hexagonal Chains

An explicit combinatorial expression is obtained for the Zhang-Zhang polynomial (also known as “Clar cover polynomial”) of a large class of pericondensed benzenoid systems, the multiple linear hexagonal chains Mn,m. By means of this result, various problems encountered in the Clar theory of Mn,m are also resolved: counting of Clar and Kekulé structures, determining the Clar number, and calculating the sextet polynomial.